ARTICLE IN PRESS

Applied and Computational Mechanics 11 (2017) XXX–YYY

Determination of group velocity of propagation of Lamb waves inaluminium plate using piezoelectric transducers

Z. Lasovaa,∗, R. Zemcıka

aFaculty of Applied Sciences, University of West Bohemia, Univerzitnı 8, 306 14 Plzen, Czech Republic

Received 2 March 2017; accepted 2 June 2017

Abstract

A prior knowledge of group velocities of Lamb wave modes is a key for analysis of time signals in guided-wave based structural health monitoring. The identification of multiple wave modes may be complicated due todependency of group velocity on frequency (dispersion). These dependencies for infinite plate of constant thicknesscan be calculated by a numerical solution of analytic equation. Two alternative approaches to determine groupvelocities of zero-order Lamb wave modes in aluminum plate were used in this work: Two-dimensional FastFourier Transform (2D-FFT) and methods of time-frequency processing. 2D-FFT requires a high number of timesignals in equidistant points, therefore it was applied on data from finite element analysis of wave propagation inthe plate. Group velocities for chosen frequencies were also determined using wavelet transform (WT) of signalsas differencies of times of arrival measured by a pair of piezoelectric transducers. The results from 2D-FFT andwavelet transform were compared to the analytic solution.c© 2017 University of West Bohemia. All rights reserved.

Keywords: Lamb waves, piezoelectric transducers, 2D-FFT, wavelet transform

1. Introduction

Lamb waves attracted scientific interest in recent years due to their potential use in the structuralhealth monitoring (SHM, automated structural defect detection). They are a type of elastic wavespropagating in plates or hollow cylinders, so Lamb waves-based SHM can be successfully usedfor shell structural parts (such as aircraft fuselage [3], pressure vessels, wind turbine blades etc.),or for monitoring of pipelines.

Fig. 1. Coordinate system of the plate

Considering Lamb waves in plates, plane of particle motion is defined by the direction ofpropagation and the normal to the plate (e.g. plane x-z in Fig. 1 with x as the direction ofpropagation). In contrast to bulk waves existing in two wave modes (longitudinal, transverse

∗Corresponding author. Tel.: +420 377 632 327, e-mail: zlasova@kme.zcu.cz.https://doi.org/10.24132/acm.2017.346

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shear), Lamb waves have generally infinite number of modes. These modes are divided totwo groups: symmetric (Si) and antisymmetric (Ai), with respect to the middle plane. It isadvantageous to use low frequency range, where only two zero-order modes exist (denotedas S0 and A0). With increasing frequency higher modes occur. The determination of groupvelocities of the Lamb modes is the key for identification of waves reflected by a damage [13]or to locate the sources of acoustic emission [12].

Lamb waves are dispersive, which means their group and phase velocities depend onfrequency. These dependencies for each Lamb wave mode are called dispersion curves andcan be calculated by numerical solution of Rayleigh-Lamb equation [5]

tan βd

tanαd= −

[4αβk2

(k2 − β2)

]±1

, (1)

where the exponent+1 refers to the symmetric modes and −1 to the antisymmetric modes. Thecoefficients α, β are defined by

α =

√ω2

c2L− k2 and β =

√ω2

c2T− k2. (2)

In equations (1) and (2), ω is angular frequency, k is wavenumber, d is half-thickness of theplate, cL is velocity of longitudinal wave and cT is velocity of transverse shear wave.

Rayleigh-Lamb equation applies on theoretical model of infinite isotropic plate of constantthickness. An alternative method of approximation of dispersion curves applicable on anyisotropic thin-walled structure was developed: two-dimensional Fast Fourier Transform (2D-FFT) presented by Alleyne and Cawley [1]. The input data for the 2D-FFT are time-records inmultiple positions along the direction of propagation of the wave. Input data can be collectedby measurement (by moving laser [4, 10] or air-coupled ultrasonic probe [6]) or by numericalsolution [2, 8]. In this work, finite element model of the cross-section of the plate was used tocollect data for 2D-FFT.

The dispersion curves for group velocities can be also determined for given frequenciesby direct measurement of time of arrival of single frequency components. In this case it isadvantageous to excite waves, in which this frequency component dominates. This can beachieved by choice of proper actuating signal. From the signals collected by sensors, thesignificant frequency component is separated by means of signal processing methods. WaveletTransform (WT) is preferred by most of authors [9,13], Wigner-Ville Distribution [14] or Short-Time Fourier Transform [15] were also proposed. In this case wavelet transform was chosen,because it provides best determination of time-of-arrival of acoustic event among mentionedmethods. This is thanks to variable size of window function (wavelet) and similarity of waveletfunction with actuated sine burst.

In this paper, dispersion curves were calculated by 2D-FFT of numerical results. Groupvelocities for a set of input frequencies in range 100–500 kHz were determined by time-frequency analysis of signals from piezoelectric transducers. Calculated and measured groupvelocities were compared with the analytical dispersion curves.

2. Determination of dispersion curves by 2D-FFT

Due to the fact that Lamb waves are periodical both in temporal and spatial domain, twodimensional Fast Fourier Transform (2D-FFT) can be used for transformation of time-spatial

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data to wavenumber-frequency domain. If x is the direction of propagation of Lamb waves,2D-FFT is defined as

H(k, f) =∫ ∞

−∞

∫ ∞

−∞u(x, t)e−i(kx+t) dx dt, (3)

where f is frequency. The matrix u(x, t) is composed of time responses (displacements ux, uz)in equidistant points along x. To achieve sufficient spatial resolution, a high number of these“sensors” must be used.

Finite element model of the aluminum plate was created to collect the data for 2D-FFT. Themodel was a cross-section x, z of the plate and plane strain was assumed, which correspondsto the character of Lamb waves. The thickness 2d = 2 mm corresponded to an experimentalplate used later and the finite elements were four-node quadrilaterals of size 0.25 mm. Materialparameters of the plate are presented in Table 1. Static material parameters were used — tensilemodulus and Poisson’s ratio measured by a tensile test of set of 3 specimens cut from the plate.

Table 1. Material parametrs of the aluminum plate

Tensile modulus E [GPa] 69.2Poisson’s ratio ν [–] 0.287Density ρ [kg ·m−3] 2 600Longitudinal wave velocity cL [m · s−1] 5 884Shear wave velocity cS [m · s−1] 3 216

Abaqus Explicit v.6-14 was used for the simulation of wave propagation. The time incrementwas set toΔt = 4.0×10−8 s. The model was loaded at one end by a time-dependent displacement(Hann) pulse with duration 2.0 × 10−6 s (presented in Fig. 2). Its frequency spectrum wassufficient to excite broadband frequency response.

Fig. 2. Exciting pulse and its frequency spectrum (normalised amplitudes)

It turned out to be advantageous to excite the zero-order symmetric and antisymmetric modesseparately. To excite unique mode, displacement-based load across the thickness was used. Themagnitudes of the displacement were determined by solution of Rayleigh-Lamb equation. Onlythe dominant displacement directions were used as boundary conditions for each mode:

u0z(z) = Aksin βz

cosβh+ 2

αβ sinαz

(k2 − β2) cosαh(4)

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in the case of symmetric mode S0 and

u0x(z) = Aiβsin βz

sin βh− 2k2 sinαz

(k2 − β2) sinαh(5)

for antisymmetric mode A0.

Fig. 3. Deformation of cross-section of the plate for case of symmetric and antisymmetric modes

The deformation of cross-section of the plate for both cases is presented in Fig. 3. Timehistories of displacements at 4 096 nodes on the top surface of the plate were ordered ascolumns of the matrix u(x, t). Longitudinal displacement ux was used for determination ofsymmetric modes and transversal displacement uz for antisymmetric modes. In both cases, thematrix u(x, t) was transformed by 2D-FFT to wavenumber-frequency domain and the resultingmatrices H(k, f) are visualised as colour plots in Fig. 4a–b. Elements with higher intensities(darker shade of grey) indicate admissible combinations of frequency and wavenumber, i.e. theyapproximately show the dispersion curves in k-f domain. In given frequency range 0–1 MHz,zero-order symmetric mode S0 and antisymmetric modes A0 and A1 are clearly visible. Forcomparison, theoretical dispersion curves (calculated by DCTool, a program for numericalsolution of Rayleigh-Lamb equation [7]) are superimposed to the 2D-FFT results, as is shown inFig. 4c–d. It is apparent that numerical results correspond very well with the analytic solution.

2D-FFT was also used for determination of group velocities. The group velocity generallydefined by

cg(n) =∂ω

∂k(6)

was calculated as a ratio of differences

cg(n) =ωn+1 − ωn−1

kn+1 − kn−1. (7)

The values of group velocities were determined for selected frequencies in range100–500 kHz (to ensure the presence of zero-order modes only) with step of 10 kHz. Thewavenumbers were found from the maximum values in rows of matrix H(k, f) correspondingto closest values of the given frequencies. The calculated group velocities are presented alongwith experimental data in Fig. 9 in the next section.

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Fig. 4. Result of 2D-FFT and comparison with analytic solution: a) symmetric modes by 2D-FFT,b) antisymmetric modes by 2D-FFT, c) analytic dispersion curves of symmetric modes and d) analyticdispersion curves of antisymmetric modes superimposed on 2D-FFT

3. Measurement of group velocities

The measurement of Lamb waves group velocities was performed on a plate with dimensions800×800×2mm, which was hanged on a frame by two elastic strings (as can be seen in Fig. 5)

Fig. 5. Measurement set-up: plate hanged on a frame with applied piezoelectric patches

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and secured in vertical position by a clip in the top left corner to minimize contact with the strings.Three transducers were applied on the plate, an actuator and a pair of sensors placed along theinvestigated direction of propagation. The transducers were piezoelectric patches, specificallyDuraAct P-876.SP1. Piezoceramic layer in these patches has dimensions 10× 10× 0.2 mm andit is sealed in a protective plastic foil with dimensions 16×13×0.5mm. More specifications ofthe transducers can be found in producer’s data sheet [16]. The actuator was placed in the centreof the plate. The initial setup of sensors was d1 = 150 mm and d2 = 250 mm measured fromthe actuator, but it had to be changed to d1 = 100 mm and d2 = 200 mm for some frequenciesto avoid overlay of incident and reflected wave packets. Therefore the transducers were bondedto the plate by double-side adhesive tape, which allows their relocation without any damage ofthe transducers or the plate. On the other hand, this way of attachment can impair quality ofrecorded signals.

Excitation and recording of signals from piezoelectric transducers was preformed usingAcellent ScanSentry, which is specialized device for Lamb wave-based structural health moni-toring. It operated with sampling frequency 12MHz both for actuator and sensors signals. Thedata acquisition is one-channel, therefore the measurement from the pair of collocated sensorswas sequential.

For the measurement of group velocity of certain frequency, it is necessary to isolate thisfrequency component from recorded signals. It is easier if wave packets with this frequencycarry the largest amount of energy. This could be achieved by a choice of an actuation pulse inform of sine burst with this central frequency. It is preferred to reduce influence of geometricdispersion on wave packets, as they are distorted to longer time span and their amplitudesdecrease significantly. One possibility is to use actuation pulse limited by window function,e.g. multiplied by Hann window. As presented in [11], higher number of sine periods alsoreduces the effect of dispersion, on the other hand the pulse lasts a longer period in time andthe wave packets may merge together in time signals. Hann-windowed 5-period sine burst (aspresented in Fig. 6) was chosen for all measurements. The central frequencies were set in rangeof 100–500 kHz with step by 50 kHz. Measurements were repeated 3-times for each frequencywith very low variation of results.

Fig. 6. Actuating pulse: Hann-windowed 5-sine burst, the central frequency in this example is 200 kHz

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Fig. 7. Measured voltage signals from a pair of sensors and time history of frequency componentf = 200 kHz

The group velocities were determined from the difference of arrival times detected by thepair of sensors in mutual distance 100 mm. To determine time of arrival of the wave modes, timehistory of energy of significant frequency has to be separated. The temporal signals from bothsensors (Fig. 7a–b) were processed by wavelet transform and the time history of the frequencycomponent (Fig. 7c) was extracted as a corresponding row in the matrix of scalogram (absolutevalue of the result of wavelet transform). Peaks of wavelet transform coefficient indicate apassage of forward and reflected wave modes.

The propagation of zero-order modes in time and distance from actuator is presented inFig. 8. The group velocities were calculated from time of arrivals of incident wave packets. Itis quite easy to identify the fastest forward S0 mode, as it is the first significant peak in thesignal, followed by forward A0. The dotted lines connecting first two peaks in time signals wereprolonged to the edge of plate to identify wave packets reflected from the edge. Especially inlower frequencies, forward mode A0 was overlaid by mode S0 reflected from the edge and forthese measurements the sensors were moved to positions d1 = 100 mm and d2 = 200 mm.

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Fig. 8. Propagation of zero-order wave modes through plate in time and travelled distance

The measured group velocities were compared to the results of 2D-FFT and analytic solution,as is shown in Fig. 9. The group velocities from 2D-FFT data were calculated using relation(7) and extrapolated by a polynomial. Results shows good precision in case of mode A0 modeand some deviations in case of mode S0. These errors are caused mainly by precision in spatialdomain, which is given by number of evaluated nodes of the finite element model.

Fig. 9. Dependency of group velocity on frequency of zero-order modes calculated by analytic solution,FEM + 2D-FFT and from measured data + wavelet transform

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Experimental results oscillate around theoretical dispersion curve with average difference3.4% in case of S0 and 6.4% for mode A0. The reason for this is that the mode S0 is not verydispersive in this frequency range and it can be easily found in time signals. On the other sideA0 is more dispersive in low frequency range, which is the likely source of errors. The influenceof dispersion decreases with increasing frequency.

4. Conclusion

Dispersion curves (group velocity dependent on frequency) of two zero-order Lamb wave modesin aluminum plate were obtained by two methods: two-dimensional Fourier transform appliedon data from the finite element analysis and measurement using piezoelectric transducers.

2D-FFT provides well approximated dispersion curves in k-f domain, which correlate withdispersion curves calculated by solution of Rayleigh-Lamb equation. The precision of calculationof group velocity depends on number of “sensors” and time increment of input data, i.e. samplesfor Fast Fourier Transform. In this case sufficient precision was achieved using 4 096 samples.

Measurement of group velocities by piezoelectric transducers and signal processing bywavelet transform proves to be more accurate in frequencies with lower effect of geometricdispersion. In case of symmetric mode S0 it oscillates around analytic curve with averagedifference 3.4% and 6.4% in case of anti-symmetric mode A0.

Both proposed methods showed sufficient precision for identification of zero-order Lambwave modes in time signals by determination of their group velocity. Their main advantage isapplicability on a wider set of structures in comparison with analytical solution.

Acknowledgements

This publication was supported by the project LO1506 of the Czech Ministry of Education,Youth and Sports.

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